Abstract

This research explores the structural differences between solvable and non-solvable groups within modern algebra and examines their computational implications in algorithmic group theory. Solvable groups exhibit hierarchical decomposition through abelian normal subgroups, making them computationally tractable, whereas non-solvable groups present higher complexity and resistance to decomposition. The study integrates theoretical group properties with computational experimentation to evaluate algorithmic efficiency in group recognition, word problems, and automorphism computations. A sample of 100 computational cases involving finite groups of varying complexity was analyzed. Results indicate that solvable groups significantly reduce computational complexity in algorithmic tasks compared to non-solvable groups. The study contributes to both abstract algebra and computational group theory by bridging structural theory with algorithmic performance.